"He must then respectfully make offerings to the guest-monks, in accord with the Dharma" (BNS)-- Here and in a few other places, it seems to mean the same thing as dana-- donations of money or materials to monks.

"Hence, if he does not make offerings to the sutras and moral codes, in accordance with the Dharma, he commits a secondary offense." (BNS) -- Not sure what they mean here. You can't give a lunch to a sutra. This means something different than in the 1st case. Is this just rhetorical flourish that means one should read and follow the sutras and moral codes?

Also, are offerings a part of maintaining a home altar in Chinese Buddhism? What, if any thing, is typically "offered"? What does one do with the offerings afterwards (particularly the food and the like)?

This is not necessarily limited to Chinese Buddhism either. Certain sutras, such as the Medicine Buddha sutra, have guidelines on how offerings are to be made: such as storing in a high place and hanging 5-colored flags.